We present a constructive description of monogenic functions that take values in a commutative biharmonic algebra by using analytic functions of complex variables. We establish an isomorphism between algebras of monogenic functions defined in different biharmonic planes. It is proved that every biharmonic function in a bounded simply connected domain is the first component of a certain monogenic function defined in the corresponding domain of a biharmonic plane.
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I. P. Mel’nichenko, “Biharmonic bases in algebras of second rank,” Ukr. Mat. Zh., 36, No. 2, 252–254 (1986).
V. F. Kovalev and I. P. Mel’nichenko, “Biharmonic functions on a biharmonic plane,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 8, 25–27 (1981).
I. P. Mel’nichenko and S. A. Plaksa, Commutative Algebras and Space Potential Fields [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2008).
V. F. Kovalev, Biharmonic Schwartz Problem [in Russian], Preprint No. 86.16, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1986).
N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity [in Russian], Nauka, Moscow (1986).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 12, pp. 1587–1596, December, 2009.
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Grishchuk, S.V., Plaksa, S.A. Monogenic functions in a biharmonic algebra. Ukr Math J 61, 1865–1876 (2009). https://doi.org/10.1007/s11253-010-0319-5
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DOI: https://doi.org/10.1007/s11253-010-0319-5